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1 11 (7) Let A = {- :n ɛ Z and n >1}= Let 2'3'4' ... o ifr € A S(z) = |1 if z¢ A° In this question we will prove that f is integrable on [0, 1]. (a) Prove that (f) = 1. (b) Prove that for every positive integer n, and for every e > 0, there exists a partition P of [0, 1] such that Lp(f) > 1 – - Note: this question involves two arbitrary values, namely, n and e. So you might want to first pick a value for n, say n = 2, and show that for any e > 0, there exists a partition P of [0, 1]

1 11 (7) Let A = {- :n ɛ Z and n >1}= Let 2'3'4' ... o ifr € A S(z) = |1 if z¢ A° In this question we will prove that f is integrable on [0, 1]. (a) Prove that (f) = 1. (b) Prove that for every positive integer n, and for every e > 0, there exists a partition P of [0, 1] such that Lp(f) > 1 – - Note: this question involves two arbitrary values, namely, n and e. So you might want to first pick a value for n, say n = 2, and show that for any e > 0, there exists a partition P of [0, 1]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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such that Lp(g) >1- might make answering this part easier. (c) Prove that for every e > 0, there exists a partition P of [0, 1] such that Lp(f) >1-e. Hint. Use part (b). (d) Show that I(f) = 1, and conclude that f is integrable on [0, 1]. E = 2 - €. Do a similar proof for n = 3 and n = 4. Doing this
Transcribed Image Text:such that Lp(g) >1- might make answering this part easier. (c) Prove that for every e > 0, there exists a partition P of [0, 1] such that Lp(f) >1-e. Hint. Use part (b). (d) Show that I(f) = 1, and conclude that f is integrable on [0, 1]. E = 2 - €. Do a similar proof for n = 3 and n = 4. Doing this
1 1 1 2'3'4 (7) Let A = :n e Z and n >1 Let .... So if æ ¢ A if x € A f(x) = In this question we will prove that f is integrable on [0, 1]. (a) Prove that (S) = 1. (b) Prove that for every positive integer n, and for every e > 0, there exists a partition P of [0, 1] such that Lp(f) >1 - - Note: this question involves two arbitrary values, namely, n and e. So you might want to first pick a value for n, say n = 2, and show that for any e > 0, there exists a partition P of [0, 1]
Transcribed Image Text:1 1 1 2'3'4 (7) Let A = :n e Z and n >1 Let .... So if æ ¢ A if x € A f(x) = In this question we will prove that f is integrable on [0, 1]. (a) Prove that (S) = 1. (b) Prove that for every positive integer n, and for every e > 0, there exists a partition P of [0, 1] such that Lp(f) >1 - - Note: this question involves two arbitrary values, namely, n and e. So you might want to first pick a value for n, say n = 2, and show that for any e > 0, there exists a partition P of [0, 1]
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